factorize ((-3(x+h)^3 +9(x+h)+3) - (-3x^3+9x+3) ) / h

Let's factorize the expression step by step:

The expression is:

\[
\frac{{-3(x+h)^3 + 9(x+h) + 3 - (-3x^3 + 9x + 3)}}{h}
\]

### Step 1: Expand \((-3(x+h)^3 + 9(x+h) + 3)\)
We need to expand \((-3(x+h)^3)\) and \(9(x+h)\).

Using the binomial expansion for \((x+h)^3\):

\[
(x+h)^3 = x^3 + 3x^2h + 3xh^2 + h^3
\]

Now, distribute the \(-3\):

\[
-3(x+h)^3 = -3x^3 - 9x^2h - 9xh^2 - 3h^3
\]

Next, distribute the 9 in \(9(x+h)\):

\[
9(x+h) = 9x + 9h
\]

So, expanding the whole first term:

\[
-3(x+h)^3 + 9(x+h) + 3 = (-3x^3 - 9x^2h - 9xh^2 - 3h^3) + (9x + 9h) + 3
\]

### Step 2: Simplify the numerator
Now let's subtract \((-3x^3 + 9x + 3)\) from the expanded expression:

\[
(-3x^3 - 9x^2h - 9xh^2 - 3h^3 + 9x + 9h + 3) - (-3x^3 + 9x + 3)
\]

Simplifying:

\[
= -3x^3 - 9x^2h - 9xh^2 - 3h^3 + 9x + 9h + 3 + 3x^3 - 9x - 3
\]

Canceling out the like terms:

\[
= (-9x^2h - 9xh^2 - 3h^3 + 9h)
\]

### Step 3: Factor the numerator
Now, factor \(h\) from all the terms in the numerator:

\[
= h(-9x^2 - 9xh - 3h^2 + 9)
\]

### Step 4: Simplify the expression
Now, the expression becomes:

\[
\frac{h(-9x^2 - 9xh - 3h^2 + 9)}{h}
\]

Since \(h\) appears in both the numerator and denominator, we can cancel it out:

\[
= -9x^2 - 9xh - 3h^2 + 9
\]

This is the simplified and factorized form of the expression.

Would you like more details on any step, or do you have any questions? 

Here are five related questions to explore further:

1. How does binomial expansion work for higher powers?
2. What are the common techniques for factoring polynomial expressions?
3. Can we apply this factoring process to non-cubic polynomials?
4. What if \(h\) approaches 0—how does this relate to the derivative?
5. How does the chain rule apply to this type of expression in calculus?

**Tip:** When simplifying expressions, always look for common factors to reduce complexity and avoid unnecessary steps.

Explore the factorization of the polynomial expression ((-3(x+h)^3 +9(x+h)+3) - (-3x^3+9x+3)) / h, using binomial expansion and algebraic simplification techniques. Detailed step-by-step explanation for students in advanced algebra and pre-calculus courses.

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